Well, some of my first projects were :
A knitted bookmark
A knitted and felted coaster
A garter stitch shawl
A dishcloth (although this is not very impressive looking, admittedly)
child's hand puppet
All of these can be found on the http://www.knittingpatterncentral.com
site... I can link to the individual patterns if you are interested.
If I could go back in time, I would knit that super cute drop stitch scarf that is linked to on this site (and I believe there's a whole video on how to do it)... a SUPER fast project, and it looks wonderful and cool when it's done!
Also, anything felted is wonderful because any mistakes are hidden in the felting (shrinking) process! I made a coaster and felted it the other day, it looks really neat and took me around 1-2 hours. Maybe you could do a purse, if you're feeling ambitious...